Deformed relativistic kinematics on curved spacetime -- a geometric approach

TL;DR

This paper develops a geometric framework for extending deformed relativistic kinematics from flat to curved spacetimes, ensuring consistent particle motion and connecting to non-commutative geometry.

Contribution

It introduces a systematic method to lift momentum space metrics to curved spacetimes using cotangent bundle geometry, clarifying conditions for geometric consistency.

Findings

Momentum space metrics can be lifted if dispersion relations are homogeneous or satisfy symmetry constraints.

The symmetry constraint requires invariance under local Lorentz transformations in a suitable basis.

The framework links deformed kinematics to non-commutative spacetime structures.

Abstract

Deformed relativistic kinematics have been considered as a way to capture residual effects of quantum gravity. It has been shown that they can be understood geometrically in terms of a curved momentum space on a flat spacetime. In this article we present a systematic analysis under which conditions and how deformed relativistic kinematics, encoded in a momentum space metric on flat spacetime, can be lifted to curved spacetimes in terms of a self-consistent cotangent bundle geometry, which leads to purely geometric, geodesic motion of freely falling point particles. We comment how this construction is connected to, and offers a new perspective on, non-commutative spacetimes. From geometric consistency conditions we find that momentum space metrics can be consistently lifted to curved spacetimes if they either lead to a dispersion relation which is homogeneous in the momenta, or, if they satisfy a specific symmetry constraint. The latter is relevant for the momentum space metrics encoding the most studied deformed relativistic kinematics. For these, the constraint can only be satisfied in a momentum space basis in which the momentum space metric is invariant under linear local Lorentz transformations. We discuss how this result can be interpreted and the consequences of relaxing some conditions and principles of the construction from which we started.